Cosmological models constrained by homogeneous and isotropic expansion and a phenomenological equation of state

In this manuscript, we investigate Einstein's equations for matter satisfying a phenomenological equation of state under the hypothesis the universe is expanding homogeneously and isotropically. This gives rise to a general scheme for constructing initial value problems of cosmological interest. We restrict ourselves to the spatially periodic case so that the cosmological models arising therein are naturally interpreted as spatially homogeneous and isotropic on large scales, as demanded by a weak version of the Cosmological Principle. We show the well-posedness and applicability of such a scheme for a class of spacetimes filled with generic homogeneous perfect fluids and inhomogeneous matter with bulk viscosity. We prove the existence, uniqueness, and relative stability of the solutions, as well as an additional inequality for the energy density. We establish the large-scale energy density arising through spatial averages and discuss its interpretation and applicability to Cosmology.